Harmonic Mean

Harmonic mean is a type of numerical average that is usually used in situations when the average rate or rate of change needs to be calculated. It is one of the three Pythagorean means. The remaining two are the arithmetic mean and the geometric mean. These three means or averages are very important as they see widespread use in the field of geometry and music. If we are given a data series or a set of observations then the harmonic mean can be defined as the reciprocal of the average of the reciprocal terms. i.e., it is the reciprocal of the arithmetic mean of the reciprocals.

In this article, we will take a detailed look at the different aspects of the harmonic mean (HM) along with the relationship between all these three means (AM, GM, HM).

The harmonic mean is a measure of central tendency. Say we want to determine a single value that can be used the describe the behavior of data around a central value. Then such a value is known as a measure of central tendency. In statistics, there are three measures of central tendency. These are the mean, median, and mode. The mean can be further classified into arithmetic mean, geometric mean, and harmonic mean.

Harmonic Mean Definition

The harmonic mean is a type of Pythagorean mean. To find it, we divide the number of terms in a data series by the sum of all the reciprocal terms. It will always be the lowest as compared to the geometric and arithmetic mean.

Harmonic Mean Example

Suppose we have a sequence given by 1, 3, 5, 7. The difference between each term is 2. This forms an arithmetic progression. To find the harmonic mean, we take the reciprocal of these terms. This is given as 1, 1/3, 1/5, 1/7 (the sequence forms a harmonic progression). Next, we divide the total number of terms (4) by the sum of the terms (1 + 1/3 + 1/5 + 1/7). Thus, the harmonic mean = 4 / (1 + 1/3 + 1/5 + 1/7) = 2.3864.

If we have a set of observations given by x₁, x₂, x₃....x_n. The reciprocal terms of this data set will be 1/x₁, 1/x₂, 1/x₃....1/x_n. Thus, the harmonic mean formula is given by

• HM = n / [1/x₁ + 1/x₂ + 1/x₃ + ... + 1/x_n]

Here, the total number of observations is divided by the sum of reciprocals of all observations.

The formula is shown in sigma notation as well.

Harmonic Mean of Two Numbers

Say we want to find the harmonic mean of any two numbers, a and b, in a data set. Both a and b are non-zero numbers. Thus, using the aforementioned formula we get,

n = 2

HM = 2 / [1/a + 1/b]

HM = (2ab) / (a + b)

We can follow the steps given below to find the harmonic mean of the terms in a particular observation set.

• Step 1: Take the reciprocal of each term in the given data set.
• Step 2: Count the total number of terms in the given data set. This will be n.
• Step 3: Add all the reciprocal terms.
• Step 4: Divide the value obtained in step 2 by the value from step 3. The resultant will give us the harmonic mean of the required number of terms.

Both harmonic mean and geometric mean are measures of central tendencies. Here are the differences between these two means:

Both types of means fall under the category of Pythagorean means. The table for the difference between harmonic mean vs arithmetic mean is given below.

The products of the harmonic mean (HM) and the arithmetic mean (AM) will always be equal to the square of the geometric mean (GM) of the given data set. To understand the relationship between the AM, GM, and GM we will take the help of the formulas. Say we have 2 numbers a and b.

n = 2

According to definition

AM = (a + b) / 2.

HM = 2ab / (a + b) or (ab) [2 / (a + b)]

GM = √(ab).

Now taking the square we get GM² = (ab). Using this value,

HM = GM² . [2 / (a + b)]

HM = GM² / AM.

Thus, we get

GM² = HM × AM.

Also, HM ≤ GM ≤ AM.

Note the following:

• Arithmetic mean is used when the data values have the same units.
• The geometric mean is used when the data set values have differing units.
• When the values are expressed in rates we use harmonic mean.

It is a mathematical mean that is usually used to find the average of variables when they are expressed as a ratio of different measuring units. Given below are the merits and demerits of harmonic mean:

Merits of Harmonic Mean

It is completely based on observations and is very useful in averaging certain types of rates. Other merits of the harmonic mean are given below.

• As its value remains fixed thus, it is rigidly defined.
• Even if there is a sample fluctuation, it does not get significantly affected.
• All items of the series are required to determine the harmonic mean.

Demerits of Harmonic Mean

To calculate this mean, all elements of the series must be known. In case of unknown elements, we cannot determine the harmonic mean. Given below are other demerits of harmonic mean.

• The method to calculate the harmonic mean can be lengthy and complicated.
• If any term of the given series is 0 then this mean cannot be calculated.
• The extreme values in a series greatly affect the harmonic mean.

An important property of harmonic mean is that without taking a common denominator it can be used to find multiplicative and divisor relationships between fractions. This can be a very helpful tool in industries such as finance. Given below are some other real-life applications of harmonic mean.

• It can be used to determine the patterns in the Fibonacci series.
• It is used in finance when average multiples have to be evaluated.
• It can be used to calculate quantities such as speed. This is because speed is expressed as a ratio of two measuring units such as km/hr.
• It can also be used to find the average of rates as it assigns equal weight to all data points in a sample.

Weighted harmonic mean is used when we want to find the average of a set of observations such that equal weight is given to each data point. Let x₁, x₂, x₃....x_n be the set of observations and w₁, w₂, w₃....w_n be the corresponding weights. Then the formula for weighted harmonic mean is given as follows:

Weighted HM = \(\frac{\sum_{i = 1}^{n}w_{i}}{\sum_{i = 1}^{n}\frac{w_{i}}{x_{i}}}\)

If we have normalized weights then all weights sum up to 1. That is, w₁ + w₂ + w₃ +....+ w_n = 1

Suppose we have a frequency distribution with n items x₁, x₂, x₃....x_n having corresponding frequencies f₁, f₂, f₃....f_n then the weighted harmonic mean is give as:

Weighted HM = \(\frac{n}{\sum_{i=1}^{n}\frac{f_{i}}{x_{i}}}\)

Important Notes on Harmonic Mean

• The harmonic mean is used when we want to find the reciprocal of the average of the reciprocal terms in a series.
• The formula to determine harmonic mean is n / [1/x₁ + 1/x₂ + 1/x₃ + ... + 1/x_n].
• The relationship between HM, GM, and AM is GM² = HM × AM.
• HM will have the lowest value, geometric mean will have the middle value and arithmetic mean will have the highest value.

☛Related Articles:

• AM GM HM
• Mean Mode Median
• Harmonic Mean Calculator

What is Harmonic Mean in Statistics?

When we take the reciprocal of the arithmetic mean of the reciprocal terms in a data set we get the harmonic mean. Furthermore, if there are certain weights associated with each observation then we can calculate the weighted harmonic mean.

What is Harmonic Mean Formula?

The harmonic mean is one of the measures of central tendency and is given as n / [1/x₁ + 1/x₂ + 1/x₃ + ... + 1/x_n]. Similarly the weighted harmonic mean formula is \(\frac{\sum_{i = 1}^{n}w_{i}}{\sum_{i = 1}^{n}\frac{w_{i}}{x_{i}}}\).

What is the Harmonic Mean of a and b?

If we want to find the harmonic mean of two non-zero numbers a and b then we use the general formula with n = 2. Thus, the formula for the harmonic mean of a and b is 2ab/a + b

How to Calculate Harmonic Mean?

To calculate the harmonic mean:

• We first take the sum of the reciprocals of each term in the given data set.
• Then we divide the total number of terms (n) in the data set by this value.

What is the Difference between Geometric Mean and Harmonic Mean?

When we have a data set, the geometric mean can be determined by taking the n^th root of the product of all the n terms. To find the harmonic mean (HM) we divide n by the sum of the reciprocals of the terms. The HM will always have a lower value than the geometric mean.

Is Harmonic Mean the Reciprocal of Arithmetic Mean?

Harmonic mean is not the reciprocal of the arithmetic mean. It is the reciprocal of the arithmetic mean of the reciprocal terms in a given set of observations.

What are the Merits and Demerits of Harmonic Mean?

The harmonic mean has a rigid value and does not get affected by fluctuations in the sample however if the sample contains a zero term we cannot calculate it. Also, the formula to determine the harmonic mean can result in complex computations.

What are Applications of Harmonic Mean?

Harmonic mean sees widespread use in geometry and music. It is also used to calculate the average of ratios as it equalizes all the data points.